BigDFT Formalism Daubechieswaveletsasabasissetfordensity ... · BigDFT Formalism Wavelets Algorithms Poisson Solver Features Applications Publications Si clusters Boron clusters Conclusion

BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features

ApplicationsPublications

Si clusters

Boron clusters

Conclusion

Electronic Structure Calculation Methods onAccelerators, Oak Ridge

Daubechies wavelets as a basis set for densityfunctional pseudopotential calculation

T. Deutsch, L. Genovese, D. Caliste

L_Sim – CEA Grenoble

February 7, 2012

Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese, D. Caliste

BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

A basis for nanosciences: The BigDFT project

STREP European project: BigDFT(2005-2008)Four partners, 15 contributors:CEA-INAC Grenoble (T. D.), U. Basel (S. Goedecker),U. Louvain-la-Neuve (X .Gonze), U. Kiel (R. Schneider)

Aim: To develop an ab-initio DFT codebased on Daubechies Waveletsand integrated in ABINIT (Gnu-GPL).

BigDFT 1.0 −→ January 2008

. . . Why have we done this?Test the potential advantages of a new formalism

* A lot of outcomes and interesting results

A lot can be done starting from present know-how


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Outline

1 FormalismWaveletsAlgorithmsPoisson SolverFeatures

2 ApplicationsPublicationsSi clustersBoron clusters

3 Conclusion

Next talk: Performances (MPI+OpenMP+OpenCL)


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

A DFT code based on Daubechies wavelets

BigDFT: a PSP Kohn-Sham codeA Daubechies wavelets basis has uniqueproperties for DFT usage

Systematic, Orthogonal

Localised, Adaptive

Kohn-Sham operators are analytic

Short, Separable convolutions

c̃` = ∑j aj c`−j

Peculiar numerical properties

Real space based, highly flexibleBig & inhomogeneous systems

Daubechies Wavelets

-1.5

-1

-0.5

0

0.5

1

1.5

-6 -4 -2 0 2 4 6 8

x

φ(x)

ψ(x)


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

A brief description of wavelet theory

Two kind of basis functions

A Multi-Resolution real space basisThe functions can be classified following the resolution levelthey span.

Scaling FunctionsThe functions of low resolution level are a linear combinationof high-resolution functions

= +

φ(x) =m

∑j=−m

hjφ(2x− j)

Centered on a resolution-dependent grid: φj = φ0(x− j).


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

A brief description of wavelet theory

WaveletsThey contain the DoF needed to complete the informationwhich is lacking due to the coarseness of the resolution.

= 12 + 1

2

φ(2x) =m

∑j=−m

h̃jφ(x− j) +m

∑j=−m

g̃jψ(x− j)

Increase the resolution without modifying grid spaceSF + W = same DoF of SF of higher resolution

ψ(x) =m

∑j=−m

gjφ(2x− j)

All functions have compact support, centered on grid points.


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Wavelet properties: Adaptivity

AdaptivityResolution can be refined following the grid point.

The grid is divided inLow (1 DoF) and High(8 DoF) resolution points.Points of different resolutionbelong to the same grid.Empty regions must not be“filled” with basis functions.

Localization property,real space descriptionOptimal for big & inhomogeneous systems, highly flexible


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Adaptivity of the mesh

Atomic positions (H2O), hgrid


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion


Fine grid (high resolution, frmult)


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion


Coarse grid (low resolution, crmult)


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

No integration error

Orthogonality, scaling relation∫dx φk (x)φj(x) = δkj φ(x) =

1√2

m

∑j=−m

hjφ(2x− j)

The hamiltonian-related quantities can be calculated up tomachine precision in the given basis.

The accuracy is only limited by the basis set (O(h14

grid

))

Exact evaluation of kinetic energyObtained by convolution with filters:

f (x) = ∑`

c`φ`(x) , ∇2f (x) = ∑

`

c̃` φ`(x) ,

c̃` = ∑j

cj a`−j , a` ≡∫

φ0(x)∂2xφ`(x) ,


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Separability in 3D

The 3-dim scaling basis is a tensor product decomposition of1-dim Scaling Functions/ Wavelets.

φex ,ey ,ezjx ,jy ,jz (x ,y ,z) = φ

exjx (x)φ

eyjy (y)φ

ezjz (z)

With (jx , jy , jz) the node coordinates,

φ(0)j and φ

(1)j the SF and the W respectively.

Gaussians and waveletsThe separability of the basis allows us to save computationaltime when performing scalar products with separablefunctions (e.g. gaussians):

Initial wavefunctions (input guess)

Poisson solver (tensor decomposition of 1r )

Non-local pseudopotentials


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Wavelet families used in BigDFT code

Daubechies f (x) = ∑` c`φ`(x)

Orthogonal c` =∫

dx φ`(x)f (x)

-1.5

-1

-0.5

0

0.5

1

1.5

-6 -4 -2 0 2 4 6

LEAST ASYMMETRIC DAUBECHIES-16

waveletscaling function

Used for wavefunctions,scalar products

Interpolating f (x) = ∑j fjϕj (x)

Dual to dirac deltas fj = f (j)

-1

-0.5

0

0.5

1

-4 -2 0 2 4

scaling functionwavelet

INTERPOLATING (DESLAURIERS-DUBUC)-8

Used for charge density,function products

“Magic Filter” method (A. Neelov, S. Goedecker)The passage between the two basis sets can be performedwithout losing accuracy.


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Operations performed

The SCF cycleOrbital scheme:

Hamiltonian

Preconditioner

Coefficient Scheme:

Overlap matrices

Orthogonalisation

Comput. operationsConvolutions

BLAS routines

FFT (Poisson Solver)

Why not GPUs?

Real Space Daub. Wavelets


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Kohn-Sham Equations: Computing Energies

Calculate different integrals

E[ρ] = K [ρ] + U[ρ]

K [ρ] =−22~2

me∑

i

∫V

dr ψ∗i ∇

2ψi

U[ρ] =∫

Vdr Vext(r)ρ(r)+

12

∫V

dr dr ′ρ(r)ρ(r ′)|r − r ′|︸︷︷︸

Hartree

+ Exc[ρ]︸︷︷︸exchange−correlation

We minimise with the variables ψi(r) and fi

with the constraint∫

Vdr ρ(r) = Nel .


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Direct Minimisation: Flowchart{

ψi = ∑a

c iaφa

}Basis: adaptive mesh (0, . . . ,Nφ)

Orthonormalized

ρ(r) =occ

∑i|ψi (r)|2 Fine non-adaptive mesh: < 8Nφ

inv FWT + Magic Filter

VH(r) =∫

G(.)ρ VeffectiveVxc [ρ(r)] VNL({ψi})

Poisson solver

− 12 ∇2 Kinetic Term

δc ia =− ∂Etotal

∂c∗i (a)+∑

jΛij c

ja

Λij =< ψi |H|ψj >

FWT

FWT

cnew ,ia = c i

a + hstepδc ia

Steepest Descent,DIIS

preconditioningStop when δc ia small


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion


ψi = ∑a

c iaφa


Orthonormalized

ρ(r) =occ



VH(r) =∫


Poisson solver



∂c∗i (a)+∑

jΛij c

ja


FWT

FWT

cnew ,ia = c i

a + hstepδc ia




BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion


ψi = ∑a

c iaφa


Orthonormalized

ρ(r) =occ



VH(r) =∫


Poisson solver



∂c∗i (a)+∑

jΛij c

ja


FWT

FWT

cnew ,ia = c i

a + hstepδc ia




BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion


ψi = ∑a

c iaφa


Orthonormalized

ρ(r) =occ



VH(r) =∫


Poisson solver



∂c∗i (a)+∑

jΛij c

ja


FWT

FWT

cnew ,ia = c i

a + hstepδc ia




BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion


ψi = ∑a

c iaφa


Orthonormalized

ρ(r) =occ



VH(r) =∫


Poisson solver



∂c∗i (a)+∑

jΛij c

ja


FWT

FWT

cnew ,ia = c i

a + hstepδc ia




BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion


ψi = ∑a

c iaφa


Orthonormalized

ρ(r) =occ



VH(r) =∫


Poisson solver



∂c∗i (a)+∑

jΛij c

ja


FWT

FWT

cnew ,ia = c i

a + hstepδc ia


preconditioning

Stop when δc ia small


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion


ψi = ∑a

c iaφa


Orthonormalized

ρ(r) =occ



VH(r) =∫


Poisson solver



∂c∗i (a)+∑

jΛij c

ja


FWT

FWT

cnew ,ia = c i

a + hstepδc ia




BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Poisson Solver using Green function

What is still missing is a method that can solve Poisson’sequation with free boundary conditions for arbitrary chargedensities.

V (r) =∫

ρ(r′)|r− r′|

dr′

Real Mesh (1003): 106 integral evaluations in each point!

Interpolating f (x) = ∑j fjϕj (x)

Dual to dirac deltas fj = f (j)

The expansion coefficientsare the point values on agrid.

-1

-0.5

0

0.5

1

-4 -2 0 2 4

scaling functionwavelet

INTERPOLATING (DESLAURIERS-DUBUC)-8


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Poisson Solver using interpolating scaling function

The Hartree potential is calculated in the interpolating scaling

function basis using a tensor decomposition of1r

.

Poisson solver with interpolating scaling functions

On an uniform grid it calculates: VH(~j) =∫

d~xρ(~x)

|~x−~j|4 Very fast and accurate, optimal parallelization

4 Can be used independently from the DFT code

4 Integrated quantities (energies) are easy to extract

8 Non-adaptive, needs data uncompression

Explicitly free boundary conditionsNo need to subtract supercell interactions→ charged systems.


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Poisson Solver for surface boundary conditions

A Poisson solver for surface problemsBased on a mixed reciprocal-direct space representation

Vpx ,py (z) =−4π

∫dz ′G(|~p|;z− z ′)ρpx ,py (z) ,

4 Can be applied both in real or reciprocal space codes

4 No supercell or screening functions

4 More precise than other existing approaches

Example of the plane capacitor:

Periodic

x

y

V

x

Hockney

x

y

V

x

Our approach

x

y

V

x


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Basis set features

BigDFT features in a nutshell4 Arbitrary absolute precision can be achieved

Good convergence ratio for real-space approach (O(h14))

4 Optimal usage of the degrees of freedom (adaptivity)Optimal speed for a systematic approach (less memory)

4 Hartree potential accurate for various boundaryconditionsFree and Surfaces BC Poisson Solver(present also in CP2K, ABINIT, OCTOPUS)

* Data repartition is suitable for optimal scalabilitySimple communications paradigm, multi-level parallelisationpossible (and implemented)

Improve and develop know-howOptimal for advanced DFT functionalities in HPC framework


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Optimal for isolated systems

Test case: cinchonidine molecule (44 atoms)

10-4

10-3

10-2

10-1

100

101

105 106

Abs

olut

e en

ergy

pre

cisi

on (

Ha)

Number of degrees of freedom

Ec = 125 Ha

Ec = 90 Ha

Ec = 40 Ha

h = 0.3bohr

h = 0.4bohr

Plane waves

Wavelets

Allows a systematic approach for moleculesConsiderably faster than Plane Waves codes.10 (5) times faster than ABINIT (CPMD)Charged systems can be treated explicitly with the same time


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Systematic basis set

Two parameters for tuning the basisThe grid spacing hgrid

The extension of the low resolution points crmult


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

BigDFT version 1.6.0: (ABINIT-related) capabilities

http://inac.cea.fr/L_Sim/BigDFTIsolated, surfaces and 3D-periodic boundary conditions(k-points, symmetries)

All XC functionals of the ABINIT package (libXC library)

Hybrid functionals, Fock exchange operator

Direct Minimisation and Mixing routines (metals)

Local geometry optimizations (with constraints)

External electric fields (surfaces BC)

Born-Oppenheimer MD, ESTF-IO

Vibrations

Unoccupied states

Empirical van der Waals interactions

Saddle point searches (NEB, Granot & Bear)

All these functionalities are GPU-compatible


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

The vessel: Ambitions from BigDFT experience

Reliable formalismSystematic convergence properties

Explicit environments, analytic operator expressions

State-of-the-art computational technologyData locality optimal for operator applications

Massive parallel environments

Material accelerators (GPU)

New physics can be approachedEnhanced functionalities can be applied relatively easily

Beyond the DFT approximations

Order N method (in progress)


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Production version (robustness)

1 Energy Landscape of Fullerene Materials: A Comparison of Boron to BoronNitride an Carbon, PRL 106, 2011

2 Low-energy boron fullerenes: Role of disorder and potential synthesispathways, PRB 83, 2011

3 Optimized energy landscape exploration using the ab initio basedactivation-relaxation technique, JCP 135, 2011

4 The effect of ionization on the global minima of small and medium sizedsilicon and magnesium clusters, JCP 134, 2011

5 Energy landscape of silicon systems and its description by force fields, tightbinding schemes, density functional methods, and quantum Monte Carlomethods, PRB 81, 2010

6 First-principles prediction of stable SiC cage structures and their synthesispathways, PRB 82, 2010

7 Metallofullerenes as fuel cell electrocatalysts: A theoretical investigation ofadsorbates on C59Pt, PCCP 12, 2010

8 Structural metastability of endohedral silicon fullerenes, PRB 81, 20109 Adsorption of small NaCl clusters on surfaces of silicon nanostructures,

Nanotechnology 20, 200910 Ab initio calculation of the binding energy of impurities in semiconductors:

Application to Si nanowires, PRB 81, 2010


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Metal doped silicon clusters

Are fullerene like Si20 cages ground state configurations?

Collaboration with the group of S. Goedecker, Basel University


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Ground state structures of metal doped Si clusters

Never a fullerene configuration!

Ba Ca Cr Cu K Na

Pb Rb Sr Ti V Zr

Phys. Rev. B 82 201405(R) 2010


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Is it possible to have Boron fullerenes?

Boron sheet (Ismail-Beigi) B80 (G. Szwacki)


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Lower energy B80 cage structures

A (0.00)

C (0.22)

B (-0.82)

D (-0.22)

F (-0.35)

E (-0.22)

B80

20:0

B80

8:12 B80

12:8

B80

10:10B80

14:6

B80

14:6

Phys. Rev. B 83, 081403(R) 2011


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Lowest energy B80 cluster

Not a cage!

Phys. Rev. Lett. 106, 225502 (2011)


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

The configurational energy spectrum of B80 and C60


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Conclusions

BigDFT code: a modern approach for nanosciences4 Flexible, reliable formalism (wavelet properties)

4 Easily fit with massively parallel architecture

4 Open a path toward the diffusion of Hybrid architectures

Development in progress4 Wire boundary conditions and non-orthorhombic system

(full and robust DFT code)

4 PAW pseudopotentials (smoother pseudopotentials)

4 Order N methods (localization regions): large system,QM/QM scheme for embedding systems

BigDFT tutorial on WednesdayHands-on, Basic runs, Scalability, GPUs


BigDFT

FormalismWavelets

Algorithms

Poisson Solver

Features


Si clusters

Boron clusters

Conclusion

Acknowledgments

CEA Grenoble – L_SimL. Genovese, D. Caliste, B. Videau, P. Pochet, M. Ospici,I. Duchemin, P. Boulanger, E. Machado-Charry, S. Krishnan,F. Cinquini

Basel University – Group of Stefan GoedeckerS. A. Ghazemi, A. Willand, M. Amsler, S. Mohr, A. Sadeghi,N. Dugan, H. Tran

And other groupsMontreal University:L. Béland, N. Mousseau

European Synchrotron Radiation Facility:Y. Kvashnin, A. Mirone, A. Cerioni

ABINIT community (BigDFT inside, strong reusing)


Documents

BigDFT Formalism Daubechieswaveletsasabasissetfordensity ... · BigDFT Formalism Wavelets Algorithms Poisson Solver Features Applications Publications Si clusters Boron clusters Conclusion